**Introduction to the Six Hyperbolic Functions**

There are six *hyperbolic functions* that are similar in some ways to the circular functions. They are known as the *hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant,* and*hyperbolic cotangent*. In formulas and equations, they are abbreviated sinh, cosh, tanh, csch, sech, and coth respectively.

The hyperbolic functions are based on certain characteristics of the *unit hyperbola*, which has the equation *x*^{2} - *y*^{2} = 1 in rectangular coordinates. Hyperbolic functions are used in certain engineering applications.

The circular functions operate on angles. In theory, the hyperbolic functions do too. Units are generally not mentioned for the quantities on which the hyperbolic functions operate, but they are understood to be in radians. Greek symbols are not always used to denote these variables. Plain lowercase English italicized *x* and *y* are common. Some mathematicians, scientists, and engineers prefer to use *u* and *v* . Once in a while you’ll come across a paper where the author uses the lowercase italicized Greek alpha ( *α* ) and beta ( *β* ) to represent the angles in hyperbolic functions.

**Powers Of ***e*

*e*

Once we define the hyperbolic sine and the hyperbolic cosine of a quantity, the other four hyperbolic functions can be defined, just as the circular tangent, cosecant, secant, and cotangent follow from the circular sine and cosine.

In order to clearly define what is meant by the hyperbolic sine and the hyperbolic cosine, we use *base-e exponential functions* . These revolve around a number that is denoted e. This number has some special properties. It is an *irrational number* —a number that can’t be precisely expressed as a ratio of two whole numbers. The best we can do is approximate it. (The term “irrational,” in mathematics, means “not expressible as a ratio of whole numbers.” It does not mean “unreasonable” or “crazy.”)

If you have a calculator with a function key marked “ *e ^{x}* ” you can determine the value of e to several decimal places by entering the number 1 and then hitting the “

*e*” key. If your calculator does not have an “

^{x}*e*” key, it should have a key marked “ln” which stands for

^{x}*natural logarithm*, and a key marked “inv” which stands for

*inverse*. To get e from these keys, enter the number 1, and then hit “inv” and “ln” in succession. You should get a number whose first few digits are 2.71828.

If you want to determine the value of e ^{x} for some quantity *x* other than 1, you should enter the value *x* and then hit either the “ *e ^{x}* ” key or else hit the “inv” and “ln” keys in succession, depending on the type of calculator you have. In order to find e

^{– x}, find e

^{x}first, and then find the reciprocal of this by hitting the “1/x” key.

If your calculator lacks exponential or natural logarithm functions, it is time for you to go out and buy one. Most personal computers have calculator programs that can be placed in “scientific mode,” where these functions are available.

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